A building contractor is planning to build an apartment complex with one, two or three bedroom apartments. Let x,y,z respectively denote the number of apartments of each type to be built. Suppose that the builder will spend a total of $8,277,000, and that the costs for the three types of apartments are $25,000, $35,000, and $54,000 respectively. Then the 'cost equation' for the builder is:

?x + ?y + 54 z =?

Suppose he plans to build a total of 236 apartments, then complete the equation in x, y, and z which describes this:

If we express the cost in thousands of dollars, the costs are 25, 35 and 54 K$.
So the budget is 8277 k$.
The cost equation is then:
25x + 35y + 54z = 8277 .....(1)
The total number of apartments built
x+y+z = 236 ....(1)

Solve the system of equations (1) and (2) by eliminating x:
25x+35y+54z=8277 ....(1)
25x+25y+25z =5900 ....25*(2)
Subtract:
10y+29z=2377....(3)
Therefore, let z=t, then
x=236-y-t.....(1a)
y=(2377-29t)/10.....(2a)
z=t.......(3a)
where t is an integer such that x,y,z∈ℤ+.

with the only restriction that x,y,z are integers where x≥0, y≥0, and z≥0.

Equation (3) can be solved in integers y and z and 2377 with the help of the Euclid Algorithm applied to Diophantine equations to give
y=7131+29n
z=-2377-10n
Solving for n with 236≥y≥0 and 236≥z≥0, and such that 236≥x≥0, we get
8 possible answers for (x,y,z) as follows:
(4,229,3)
(23,200,13)
(42,171,23)
(61,142,33)
(80,113,43)
(99,84,53)
(118,55,63)
(137,26,73)
Check that each solution satisfies all the given conditions (cost and number of units).
For example:
25*137+35*26+54*73=8277
137+26+73=236.

If you have not done Diophantine equations before, then the answer is provided by 1a to 3a, with an appropriate choice of t to give x,y and z as integers, and satisfies x,y,z≥0.

Obvious choices for t are 10k+3 (because the last digit will end in a 7, which makes y an integer.

If no one answers in the next while, I suggest you repost your question, preferably indicating your subject, such as algebra, number theory, diophantine euqations, etc.

However, if you had read through the first six lines, you would have found the answer to the first two of the three questions. Correction:
x+y+z = 236 ....(1)
Above equation should have read equation (2), which is the answer to the second question.

The rest is at college or university level, but unfortunately I do not know if you are doing algebra or number theory. The solution (third question) is long because there are only two equations for three unknowns, and there are eight possible answers, for example x=4, y=229, z=3, etc.

Good luck!

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